Alessandra Vizzaccaro Seminar September 14, 2021

Reduced order modelling of geometrically nonlinear large finite elements structures: Direct parametrisation of invariant manifolds via high-order order expansion

Alessandra Vizzaccaro

University of Bristol

September 14, 2021

Abstract:

The increase of flexible and lightweight structures in modern mechanical design, motivates the development of accurate, yet efficient, numerical methods which can model the geometric nonlinear behaviour of such structures as they experience large amplitude vibration. Slender structures in large amplitude vibration exhibit geometric nonlinear effects that modify their dynamics; to accurately model structures with complex geometries, large finite element models with a high number of degrees of freedom are often required. Numerically, the geometric nonlinear forces affect all the elements in the structure thus making the full-order simulation of such models very demanding in terms of computational cost. Reduced order models of geometrically nonlinear structures are then an attractive solution that can drastically reduce the size of the problem whilst maintaining the accuracy high.

Direct model order reduction methods, such as implicit condensation and quadratic manifold method, give an explicit expression directly in physical coordinates of the nonlinear mapping vectors required to build reduced models of geometrically nonlinear structures. Only few calculations, often performed non-intrusively in commercial finite element software are then necessary to construct the ROM. However, such methods are based on a priori assumptions, such as the independence of the reduction manifold from the velocity, that can make them unsuitable to certain problems.

Conversely, the proposed parametrisation method produces an asymptotic approximation of the invariant manifold, thus providing an exact reduced order model. The method introduces two nonlinear mappings, related to both displacement and velocity, in the form of a generic order polynomial expansion. The same expansion is performed on the reduced-order dynamics which is computed following three different styles of parametrisation: the graph style, the complex normal form style, and the real normal form style.

In this talk, a review of direct model order reduction methods will be given, with particular focus on the proposed method for parametrisation of invariant manifolds. The accuracy of the reduction of large FE models will be demonstrated by comparison to full-order harmonic balance simulations. An overview of the possible challenges faced by various reduced order modelling techniques will be provided by means of selected examples such as curved structures with softening-hardening behaviour, structures in large rotations with inertial nonlinearities, and internally resonant structures.

Biography:

Alessandra Vizzaccaro is a Research Associate in Dynamics and Control at the University of Bristol under the DigiTwin project. She graduated in Mechanical Engineering from La Sapienza, University of Rome and completed her PhD in Dynamics at Imperial College London. Her research focuses on model order reduction, nonlinear vibration, and real time hybrid testing.